When does x/x not equal 1?

*To*: mathgroup at smc.vnet.net*Subject*: [mg32417] When does x/x not equal 1?*From*: mjperson at MIT.EDU (Michael J Person)*Date*: Sat, 19 Jan 2002 01:17:15 -0500 (EST)*Organization*: Massachusetts Institute of Technology*Sender*: owner-wri-mathgroup at wolfram.com

Greetings all: I'm having a bit of a problem with Math 4.0.1.0 on Mac OS 9.1... Specifically we have a lot of large planetary calculations that only occasionally produce piles of complex numbers of the form: 103.3 + 3.4432 *10^-39 I These tiny complex pieces then get carried all around in our calculations and destroy the formatting of our outputs and things even though the real part seems to be a quite reasonable answer to our problems. Hunting it down piece by piece, we seem to have traced it to places where we're taking the square roots of things that include ratios of numbers that should be equaling 1 at the boundaries but aren't. Sqrt[1-x/x] -> tiny complex number instead of 0! It seem to be that x/x that's causing the problem in our codes, but as we try to track it down, all sort of mystifying behavior starts to manefest. Among the worst of it are the following sorts of things: In[1]:= (x = (1.992+$MachineEpsilon)) //FullForm Out[1]//FullForm= 1.9920000000000002` In[2]:= x/x //FullForm Out[2]//FullForm= 0.9999999999999999` Now, I'd guess that wonky things start happening when you start fooling around with numbers near $MachineEpsilon, but shouldn't x/x = 1? Worse, is when we tried to fix the problem in various ways, we came up with terribly confusing things like the following: In[3]:= x/x //FullForm Out[3]//FullForm= 0.9999999999999999` (*That's just like above.) In[4]:= Divide[x,x] //FullForm Out[4]//FullForm= 1.` Ah ha! That's just the sort of thing we want in our calculations. So we try to see what the difference is.... In[5]:= Divide[a,b] //FullForm Out[5]//FullForm= Times[a, Power[b, -1]] In[6]:= a/b //FullForm Out[6]//FullForm= Times[a, Power[b, -1]] They're handled exactly the same way! So, can anyone explain to me what's going on here, and how I can keep very small complex numbers from arising in all of those cases when x/x != 1. (As I mentioned these problems are completely intermittent. In the examples given above, they do not occur for 1.991, or 1.994, just 1.992 and 1.993. Bah!) Thanks for anything you can tell me. -Michael Person mjperson at mit.edu Massachusetts Institute of Technology Department of Earth, Atmospheric, and Planetary Sciences